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Chapter 5Probability Density Functions of L-values

To characterize complex systems such as bit-interleaved coded modulation (BICM) receivers, it is convenient to study their building blocks independently. An important building block in a BICM receiver is the demapper , whose role consists in calculating the L-values. In this chapter, we formally describe its behavior from a probabilistic point of view. The models developed here will be used in the chapters that follow, to study the performance of BICM transceivers.

This chapter is organized as follows. Section 5.1 motivates the need for finding the probability density function (PDF) of the L-values and shows its challenges. Sections 5.2 and 5.3 explain how to calculate the PDFs for 1D and 2D constellations, respectively. PDFs of the L-values in fading channels are discussed in Section 5.4 and Gaussian approximations are provided in Section 5.5.

5.1 Introduction and Motivation

5.1.1 BICM Channel

The BICM channel is defined as the entity that encompasses the interleaver, the mapper, the channel, the demapper, and the deinterleaver, as shown in Fig. 5.1. As the off-the-shelf binary encoder/decoder used in BICM operates “blindly” with respect to the channel, the BICM channel corresponds to an equivalent binary-input continuous-output (BICO) channel that separates the binary encoder and decoder.